【题目】-|||-设α1,a2,α3线性无关,证明: (alpha )_(1)+(alpha )_(2), (alpha )_(2)+(alpha )_(3), (alpha )_(3)+(alpha )_(1) 也线-|||-性无关.

步骤 1：假设线性组合
假设存在实数 $l_1, l_2, l_3$ 使得
$$l_1(\alpha_1 + \alpha_2) + l_2(\alpha_2 + \alpha_3) + l_3(\alpha_3 + \alpha_1) = 0$$
步骤 2：整理线性组合
将上述等式整理，得到
$$l_1\alpha_1 + l_1\alpha_2 + l_2\alpha_2 + l_2\alpha_3 + l_3\alpha_3 + l_3\alpha_1 = 0$$
进一步整理为
$$(l_1 + l_3)\alpha_1 + (l_1 + l_2)\alpha_2 + (l_2 + l_3)\alpha_3 = 0$$
步骤 3：利用线性无关性
由于 $\alpha_1, \alpha_2, \alpha_3$ 线性无关，上述等式成立的充要条件是
$$\left \{ \begin{matrix} l_1 + l_3 = 0 \\ l_1 + l_2 = 0 \\ l_2 + l_3 = 0 \end{matrix} \right.$$
步骤 4：求解方程组
解上述方程组，得到
$$l_1 = l_2 = l_3 = 0$$